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Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy, and the uncertainty principle and is generally used in single-body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.

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Why $2j+1$ number of states?

The $2j$ is from the positive and negative j values, and the additional $+1$ accounts for $j=0$.
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